# Translate group definition into geometry system

I need to reword the definition of group (the four axioms: closure, associativity, identity and invertibility) to be lines and points of non-Euclidean geometry (the axiom system defined as geometry). And I just can't find an analogue for points and lines :(

I.e. for example, the set $\{0, 1, 2, 3\}$ relative to the operation $+_{\pmod 4}$ could be the model for the above system, but I simply cannot understand how to "draw" it using lines and points.